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\title{MTH436 - Homework 5}
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\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 3/21/18.
 
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 \section{Warm-up Questions}
 
 \begin{que}
 Let $f \colon [0,1] \to \R$ be defined by $f(x) = x^2$. For the following tagged partitions $\Pdot$, compute $S(f;\Pdot)$.
 \begin{enumerate}
  \item $\Pdot = \left\{ \left( \left[0,\frac{1}{2} \right],0\right),\left(\left[\frac{1}{2},1\right],\frac{1}{2}\right)\right\}$
  \item $\Pdot = \left\{ \left( \left[0,\frac{1}{2} \right],\frac{1}{4} \right),\left(\left[\frac{1}{2},1\right],\frac{3}{4}\right)\right\}$
  \item $\Pdot = \left\{ \left( \left[0,\frac{1}{2} \right], \frac{1}{2}\right),\left(\left[\frac{1}{2},1\right],1\right)\right\}$ 
  \item $\Pdot = \left\{ \left( \left[\frac{k-1}{n},\frac{k}{n} \right], \frac{k-1}{n}\right) \middle|  k = 1,2,\ldots,n  \right\}$
  \item $\Pdot = \left\{ \left( \left[\frac{k-1}{n},\frac{k}{n} \right], \frac{2k-1}{2n}\right) \middle|  k = 1,2,\ldots,n  \right\}$
  \item $\Pdot = \left\{ \left( \left[\frac{k-1}{n},\frac{k}{n} \right], \frac{k}{n}\right) \middle|  k = 1,2,\ldots,n  \right\}$
 \end{enumerate}
\end{que}
 
 \begin{que}
  Suppose $f \in \Rint[a,b]$. Let $(\Pdot_n)$ be a sequence of tagged partitions such that $\| \Pdot_n \| \to 0$. Prove that $\int_a^b f = \lim_{n \to \infty} S(f;\Pdot_n)$.
 \end{que}

 \begin{que}
  Let $\Pdot = \{ (I_k = [x_{k-1},x_k], t_k) \}_{k=1}^{n}$ be a tagged partition of $[0,3]$ with $\| \Pdot \| < \delta$. Define
  \begin{align*}
   &\Pdot_1 = \{ I_k \in \mathcal{P} \mid t_k \in [0,1] \} \\
   &\Pdot_2 = \{ I_k \in \mathcal{P} \mid t_k \in [1,2] \}.
  \end{align*}
Prove the following.
\begin{enumerate}
 \item $[0,1-\delta] \subseteq \bigcup_{I_k \in \Pdot_1} I_k \subseteq [0,1+\delta]$.
 \item $[1+\delta,2-\delta] \subseteq \bigcup_{I_k \in \Pdot_2} I_k \subseteq [1-\delta,2+\delta]$.
\end{enumerate}

 \end{que}

 \begin{que}
  Suppose $f \in \Rint[a,b]$ and $|f(x)| \leq M$ for all $x \in [a,b]$. Prove that
  \[
   \left| \int_a^b f \right| \leq M(b-a) 
  \]
 \end{que}

 
 \begin{que}
  Show the Dirichlet function $f \colon [0,1] \to \R$ defined by
  \[
   f(x) = \begin{cases}
           1 \quad &\text{if $x \in \Q$}, \\
           0 \quad &\text{if $x \notin \Q$}
          \end{cases}
  \]
is not Riemann Integrable.
 \end{que}
 
 
 
 
 
 
 
 
 
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 \section{Submitted Questions}
 
 \begin{que}
  Let $f \colon [a,b] \to \R$ is bounded. Suppose there exists two sequences of tagged partitions $(\Pdot_n)$ and $(\Qdot_n)$ such that $\|\Pdot_n\| \to 0$ and $\|\Qdot_n\| \to 0$ but $\lim_{n \to \infty} S(f;\Pdot_n) \neq \lim_{n \to \infty} S(f;\Qdot_n)$. Show that $f \notin \Rint[a,b]$.
 \end{que}

 

 \begin{que}
  Let $f \colon [0,1] \to \R$ be defined by
  \[
   f(x) = \begin{cases}
           \frac{1}{n} \quad &\text{if $x = \frac{1}{n}$ for some $n \in \N$}, \\
           0 \quad &\text{otherwise}.
          \end{cases}
  \]
Is $f \in \Rint[0,1]$? Justify your answer.
 \end{que}

 
 
 
 
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 \section{Challenge Questions}
 
 \begin{que}
  Let $C[0,1]$ be the set of continuous functions on $[0,1]$. For $f,g \in C[0,1]$, define $d(f,g) = \int_0^1 |f-g|$. Prove that $d$ is a metric on $C[0,1]$\footnote{You will need to assume that $C[0,1] \subseteq \Rint[0,1]$}. Is this metric space complete?
 \end{que}
 
 \begin{que}
  Prove that $\Rint[0,1]$ and $C[0,1]$ are (real) vector spaces\footnote{Come ask me for the definition of a vector space if you haven't seen it before}. Furthermore, show that $\| f \|_1 = \int_0^1 |f|$ defines a norm\footnote{Again, you can ask me if you haven't seen the definition before} on C[0,1]. Does it define a norm on $\Rint[0,1]$?
 \end{que}

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 \end{document}